Problem: $g(x)=x^4+4x^3-48x^2+6$. On which intervals is the graph of $g$ concave down? Choose 1 answer: Choose 1 answer: (Choice A) A $(-\infty,-2)$ and $(4,\infty)$ (Choice B) B $(-4,2)$ only (Choice C) C $(-2,\infty)$ only (Choice D) D $(-\infty,-4)$ and $(2,\infty)$
Solution: We can analyze the intervals where $g$ is concave up/down by looking for the intervals where its second derivative $g''$ is positive/negative. This analysis is very similar to finding increasing/decreasing intervals, only instead of analyzing $g'$, we are analyzing $g''$. The second derivative of $g$ is $g''(x)=12(x+4)(x-2)$. $g''(x)=0$ for $x=-4,2$. Since $g''$ is a polynomial, it's defined for all real numbers. Therefore, our points of interest are $x=-4$ and $x=2$. Our points of interest divide the number line into three intervals: $\llap{-}6$ $\llap{-}5$ $\llap{-}4$ $\llap{-}3$ $\llap{-}2$ $\llap{-}1$ $0$ $1$ $2$ $3$ $4$ $5$ $(-\infty, \llap{-}4)$ $( \llap{-}4,2)$ $(2,\infty)$ Let's evaluate $g''$ at each interval to see if it's positive or negative on that interval. Interval $x$ -value $g''(x)$ Verdict $(-\infty,-4)$ $x=-5$ $g''(-5)=84>0$ $g$ is concave up $\cup$ $(-4,2)$ $x=0$ $g''(0)=-96<0$ $g$ is concave down $\cap$ $(2,\infty)$ $x=3$ $g''(3)=84>0$ $g$ is concave up $\cup$ In conclusion, the graph of $g$ is concave down over the interval $(-4,2)$ only.